Global Spherically Symmetric Classical Solution to Compressible Navier–Stokes Equations with Large Initial Data and Vacuum

In this paper we consider the non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. Since the strong nonlinearity and degeneracies of the equations due to the temperature equation and vanishing of density, there are a few results about global existence of classical solution to this model. In the paper, we obtain a global classical solution to the equations with large initial data and vacuum. Moreover, we get the uniqueness of the solution to this system without vacuum. The analysis is based on the assumption \begin{document}$ \kappa(\theta) = O(1+\theta^q) $\end{document} where \begin{document}$ q\geq0 $\end{document} and delicate energy estimates.

Global classical solutions of the full compressible Navier–Stokes equations with cylindrical or spherical symmetry

Global classical large solutions to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum

Global classical solutions to 1D full compressible Navier–Stokes equations with the Robin boundary condition on temperature

Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum

The free boundary problem of planar full compressible magnetohydrodynamic equations with large initial data is studied in this paper, when the initial density connects to vacuum smoothly. The global existence and uniqueness of classical solutions are established, and the expanding rate of the free interface is shown. Using the method of Lagrangian particle path, we derive some L ∞ estimates and weighted energy estimates, which lead to the global existence of classical solutions. The main difficulty of this problem is the degeneracy of the system near the free boundary, while previous results (cf. [ 4 , 30 ]) require that the density is bounded from below by a positive constant.

We consider the one-dimensional Navier--Stokes equations for viscous compressible and heat-conducting fluids (i.e., the full Navier--Stokes equations). Because of the stronger nonlinearity and degeneracies of the equations due to the temperature equation and vanishing of density (i.e., appearance of vacuum), respectively, there are few results until now about global existence of regular solutions to the full Navier--Stokes equations. In the paper, we get a unique global classical solution to the equations with large initial data and vacuum. Our analysis is based on some delicate energy estimates together with some new ideas which were used in our previous paper [Ding, Wen, and Zhu, J. Differential Equations, 251 (2011), pp. 1696--1725] to get the high-order estimates of the solutions. This result could be viewed as the first one on the global well-posedness of regular (classical) solutions to the Navier--Stokes equations for viscous compressible and heat-conducting fluids when initial data may be large and vacuum could appear.

Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data

As a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high‐dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density and the lower bound of the temperature . Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux and the temperature by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: if ; if . This can be viewed as the first result about the global classical large solutions of the radiation hydrodynamics model with some symmetry in the high‐dimensional space.

In this paper, we investigate the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in one space dimension. The global existence and uniqueness of strong solutions with large initial data are established when there is initial vacuum. Our result may be the first result about the global strong solution with large initial data and vacuum.

We study the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in a bounded annulus Ω of ℝ3. And a result on the existence and uniqueness of global spherically symmetric classical solutions is obtained. Here the initial data could be large and initial vacuum is allowed.

Global symmetric classical solutions of the full compressible Navier–Stokes equations with vacuum and large initial data

In this paper, we study the global well-posedness of classical solution to 2D Cauchy problem of the compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ)=ρβ with β>3, then the 2D Cauchy problem of the compressible Navier–Stokes equations on the whole space R2 admits a unique global classical solution (ρ,u) which may contain vacuums in an open set of R2. Note that the initial data can be arbitrarily large to contain vacuum states. Various weighted estimates of the density and velocity are obtained in this paper and these self-contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow.

ABSTRACTWe investigate global existence of classical solution to the three dimensional Vlasov–Poisson system with radiation damping . We prove that a small perturbation of initial datum for a global classical solution verifying certain decay conditions also launches a global solution which has sharp decay. In addition, we prove that a quasineutral initial datum launches a global classical solution.

Throughout this paper, we consider the equation\[u_t - \Delta u = e^{|\nabla u|}\]with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.